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Newton's Law of Gravitation ratio

  1. Nov 12, 2007 #1
    1. The problem statement, all variables and given/known data
    A mass M is split into two parts, m and M - m, which are then separated by a certain distance. What ratio m/M maximizes the magnitude of the gravitational force between the parts?

    2. Relevant equations

    3. The attempt at a solution
    I first just tried plugging in M - m and m in for m1 and m2 to see what I get.


    I believe that I will have to take a derivative to find the maximum, but I am confused as to what I should take the derivative with respect to.
  2. jcsd
  3. Nov 12, 2007 #2
    you should not need a derivative. look at extreme cases. if the ratio of M/m was 1000, what would the force be like? what if the ratio was smaller, say 2?
  4. Nov 12, 2007 #3
    That is true, but I need to prove that is 1/2. The only way I can think of doing that is with a derivative. Although I do see the logic in what you are saying about taking the extreme cases.
  5. Nov 12, 2007 #4
    try assigning integers to the variables M and d, since they do not change based upon different m values. also, recheck the GMm-Gm part of the equation. I got something else for if i plugged in the values like you did.
  6. Nov 13, 2007 #5
    Starting with the regular equation for the force of gravity:


    Let [tex]m_0=M-m[/tex] and [tex]m_1=m[/tex]

    Substitute into the first equation:


    To maximize [tex]F[/tex], [tex]Mm-m^2[/tex] has to be maximized.

    Take its derivative with respect to [tex]m[/tex] and set it to zero:





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